...perturbations
This is a rather strong notion of hyperbolicity; it restricts such hyperbolic system to be of first-order.
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...:
(and respectively, ) indicate summation with of the first (and respectively, the first and the last) terms.
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...points
We treat here the case of an odd number of 2N+1 collocation points. We get even in §2.2.3
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...with
348#288 correspond to Chebyshev family, 396#336 correspond to Legendre, etc.
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...utilizing
Utilizing = integration by parts in this case.
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...estimate
This should be compared with the straightforward 'familiar' bound 456#396.
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....
We note that in the previous constant coefficient case, the approximate model coincides with the differential case, hence the stability estimate was nothing but the a priori estimate for the differential equation itself.
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...exponent,
To see this, use Duhammel's principle for 586#526 where 587#527 or integrate directly.
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...sign
If a(x)>0, then (meth_ps.22) is semi-bounded (and hence stable) in the weighted 666#606-norm, with 667#607.
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The last equality should be interpreted of course in the 697#637-sense, with limited by the initial 697#637-smoothness of 807#747.
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...modes).
Either one can be carried out efficiently by the FFT.
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...have
Here we utilize the fact that the error term in Gauss quadrature (2.5.4) is proportional to an intermediate value of the 2N-th derivative, 1024#964 (- e.g. consult (Gauss_Chebyshev.rule)) in the present context the inequality follows, 1025#965.
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